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Finished knot composition

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315
Advanced/Knots/main.tex
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@ -5,37 +5,15 @@
singlenumbering
]{../../resources/ormc_handout}
\usepackage{ifthen}
\usetikzlibrary{
knots,
hobby,
decorations.pathreplacing,
shapes.geometric,
calc
}
\newif{\ifShowKnots}
\ShowKnotsfalse
%\ShowKnotstrue
% Knot debugging.
% Set to true to show knot info
% Set to true to show knot measurements.
% Only used in tikzset.tex
\newif{\ifDebugKnot}
\DebugKnottrue
%\DebugKnottrue
\DebugKnotfalse
\ifDebugKnot
\tikzset{
knot diagram/draft mode = crossings,
knot diagram/only when rendering/.style = {
show curve endpoints,
%show curve controls
}
}
\fi
\input{tikzset.tex}
% From "Why knot" by
% Problems from "Why knot"
%
% Create largest crossing number with cord
% Human knot number: how many humans do you need to make the knot?
@ -44,45 +22,10 @@
%
% Figure-8 knot: mirror without letting go
\tikzset{
knot diagram/every strand/.append style={
line width = 0.8mm,
black
},
show curve controls/.style={
postaction=decorate,
decoration={
show path construction,
curveto code={
\draw[blue, dashed]
(\tikzinputsegmentfirst) -- (\tikzinputsegmentsupporta)
node [at end, draw, solid, red, inner sep=2pt]{}
;
\draw[blue, dashed]
(\tikzinputsegmentsupportb) -- (\tikzinputsegmentlast)
node [at start, draw, solid, red, inner sep=2pt]{}
node [at end, fill, red, ellipse, inner sep=2pt]{}
;
}
}
},
show curve endpoints/.style={
postaction=decorate,
decoration={
show path construction,
curveto code={
\node [fill, blue, ellipse, inner sep=2pt] at (\tikzinputsegmentlast) {};
}
}
}
}
%\usepackage{lua-visual-debug}
\begin{document}
\begin{document}
\maketitle
<Advanced 2>
<Spring 2023>
@ -91,251 +34,9 @@
Prepared by Mark on \today
}
\section{Introduction}
\definition{}
To form a \textit{knot}, take a string, tie a knot, then join the ends. \par
You can also think of a knot as a path in three-dimensional space that doesn't intersect itself:
\vspace{2mm}
\begin{center}
\begin{minipage}[t]{0.3\textwidth}
\begin{center}
\begin{tikzpicture}[scale = 0.8, baseline=(p)]
\begin{knot}
\strand
(1,2) .. controls +(-45:1) and +(1,0) ..
(0, 0) .. controls +(-1,0) and +(-90 -45:1) ..
(-1,2);
\end{knot}
\coordinate (p) at (current bounding box.center);
\end{tikzpicture}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{0.3\textwidth}
\begin{center}
\begin{tikzpicture}[scale = 0.8, baseline=(p)]
% Knot is stupid and includes invisible handles in the tikz bounding box. This line crops the image to fix that.
\clip (-2,-1.7) rectangle + (4, 4);
\begin{knot}[
consider self intersections=true,
flip crossing = 2,
]
\strand
(1,2) .. controls +(-45:1) and +(120:-2.2) ..
(210:2) .. controls +(120:2.2) and +(60:2.2) ..
(-30:2) .. controls +(60:-2.2) and +(-90 -45:1) ..
(-1,2);
\end{knot}
\coordinate (p) at (current bounding box.center);
\end{tikzpicture}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{0.3\textwidth}
\begin{center}
\begin{tikzpicture}[scale = 0.8, baseline=(p)]
\clip (-2,-1.7) rectangle + (4, 4);
\begin{knot}[
consider self intersections=true,
flip crossing = 2,
]
\strand
(0,2) .. controls +(2.2,0) and +(120:-2.2) ..
(210:2) .. controls +(120:2.2) and +(60:2.2) ..
(-30:2) .. controls +(60:-2.2) and +(-2.2,0) ..
(0,2);
\end{knot}
\coordinate (p) at (current bounding box.center);
\end{tikzpicture}
\end{center}
\end{minipage}
\end{center}
If two knots may be deformed into each other without cutting, we say they are \textit{isomorphic}. \par
If two knots are isomorphic, they are essentially the same knot.
\definition{}
The simplest knot is the \textit{unknot}. It is show below on the left. \par
The simplest nontrivial knot is the \textit{trefoil} knot, shown to the right.
\begin{center}
\begin{minipage}[t]{0.48\textwidth}
\begin{center}
\begin{tikzpicture}[baseline=(p), scale = 0.8]
\begin{knot}
\strand
(0,2) .. controls +(1.5,0) and +(1.5,0) ..
(0, 0) .. controls +(-1.5,0) and +(-1.5,0) ..
(0,2);
\end{knot}
\coordinate (p) at (current bounding box.center);
\end{tikzpicture}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
\begin{center}
\begin{tikzpicture}[baseline=(p), scale = 0.8]
\clip (-2,-1.7) rectangle + (4, 4);
\begin{knot}[
consider self intersections=true,
flip crossing = 2,
]
\strand
(0,2) .. controls +(2.2,0) and +(120:-2.2) ..
(210:2) .. controls +(120:2.2) and +(60:2.2) ..
(-30:2) .. controls +(60:-2.2) and +(-2.2,0) ..
(0,2);
\end{knot}
\coordinate (p) at (current bounding box.center);
\end{tikzpicture}
\end{center}
\end{minipage}
\end{center}
\vfill
\pagebreak
\problem{}
Below are the only four distinct knots with only one crossing. \par
Show that no nontrivial knot can have has fewer than three crossings. \par
\hint{There are 4 such knots. What are they?}
\begin{center}
\includegraphics[width=0.8\linewidth]{images/one crossing.png}
\end{center}
\begin{solution}
Draw all four. Each is isomorphic to the unknot.
\end{solution}
\vfill
\problem{}
Show that this is the unknot. \par
A wire or an extension cord may help.
\begin{center}
\includegraphics[width=0.35\linewidth]{images/big unknot.png}
\end{center}
\definition{}
As we said before, there are many ways to draw the same knot. \par
We call each drawing a \textit{projection}. Below are four projections of the \textit{figure-eight} knot.
\vspace{2mm}
\begin{center}
\includegraphics[width=0.8\linewidth]{images/figure eight.png}
\end{center}
\vspace{2mm}
\problem{}
Convince yourself that these are equivalent.
\vfill
\pagebreak
\section{Knot Composition}
Say we have two knots $A$ and $B$.
The knot $A \boxplus B$ is created by cutting $A$ and $B$ and joining their ends:
\begin{center}
\hfill
\begin{minipage}[t]{0.15\textwidth}
\begin{center}
\includegraphics[width=\linewidth]{images/composition a.png}
$A$
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{0.13\textwidth}
\begin{center}
\includegraphics[width=\linewidth]{images/composition b.png}
$B$
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{0.3\textwidth}
\begin{center}
\includegraphics[width=\linewidth]{images/composition c.png}
$A \boxplus B$
\end{center}
\end{minipage}
\hfill~
\end{center}
We must be careful to avoid new crossings when composing knots:
\vspace{2mm}
\begin{center}
\includegraphics[width=0.45\linewidth]{images/composition d.png}
\end{center}
\vspace{2mm}
We say a knot is \textit{composite} if it can be obtained by composing two other knots. \par
We say a knot is \textit{prime} otherwise.
\problem{}
For any knot $K$, what is $K \boxplus \text{unknot}$?
\vfill
\problem{}
Use a pencil or a cord to compose the figure-eight knot with itself.
\vfill
\vfill
\pagebreak{}
\problem{}
The following knots are composite. What are their prime components? \par
Try to make them with a cord! \par
\hint{Use the table at the back of this handout to decompose the second knot.}
\begin{center}
\hfill
\includegraphics[height=30mm]{images/decompose a.png}
\hfill
\includegraphics[height=30mm]{images/decompose b.png}
\hfill~\par
\vspace{4mm}
\end{center}
\begin{solution}
The first is easy, it's the trefoil composed with itself. \par
\vspace{2mm}
The second is knot $5_2$ composed with itself. \par
Note that the \say{three-crossing figure eight} is another projection of $5_2$. \par
The figure-eight knot is NOT a part of this composition. Look closely at its crossings.
\end{solution}
\vfill
\pagebreak
\input{parts/0 intro.tex}
\input{parts/1 composition.tex}
\input{parts/table}

163
Advanced/Knots/parts/0 intro.tex Normal file
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@ -0,0 +1,163 @@
\section{Introduction}
\definition{}
To form a \textit{knot}, take a string, tie a knot, then join the ends. \par
You can also think of a knot as a path in three-dimensional space that doesn't intersect itself:
\vspace{2mm}
\begin{center}
\begin{minipage}[t]{0.3\textwidth}
\begin{center}
\begin{tikzpicture}[scale = 0.8, baseline=(p)]
\begin{knot}
\strand
(1,2) .. controls +(-45:1) and +(1,0) ..
(0, 0) .. controls +(-1,0) and +(-90 -45:1) ..
(-1,2);
\end{knot}
\coordinate (p) at (current bounding box.center);
\end{tikzpicture}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{0.3\textwidth}
\begin{center}
\begin{tikzpicture}[scale = 0.8, baseline=(p)]
% Knot is stupid and includes invisible handles in the tikz bounding box. This line crops the image to fix that.
\clip (-2,-1.7) rectangle + (4, 4);
\begin{knot}[
consider self intersections=true,
flip crossing = 2,
]
\strand
(1,2) .. controls +(-45:1) and +(120:-2.2) ..
(210:2) .. controls +(120:2.2) and +(60:2.2) ..
(-30:2) .. controls +(60:-2.2) and +(-90 -45:1) ..
(-1,2);
\end{knot}
\coordinate (p) at (current bounding box.center);
\end{tikzpicture}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{0.3\textwidth}
\begin{center}
\begin{tikzpicture}[scale = 0.8, baseline=(p)]
\clip (-2,-1.7) rectangle + (4, 4);
\begin{knot}[
consider self intersections=true,
flip crossing = 2,
]
\strand
(0,2) .. controls +(2.2,0) and +(120:-2.2) ..
(210:2) .. controls +(120:2.2) and +(60:2.2) ..
(-30:2) .. controls +(60:-2.2) and +(-2.2,0) ..
(0,2);
\end{knot}
\coordinate (p) at (current bounding box.center);
\end{tikzpicture}
\end{center}
\end{minipage}
\end{center}
If two knots may be deformed into each other without cutting, we say they are \textit{isomorphic}. \par
If two knots are isomorphic, they are essentially the same knot.
\definition{}
The simplest knot is the \textit{unknot}. It is show below on the left. \par
The simplest nontrivial knot is the \textit{trefoil} knot, shown to the right.
\begin{center}
\begin{minipage}[t]{0.48\textwidth}
\begin{center}
\begin{tikzpicture}[baseline=(p), scale = 0.8]
\begin{knot}
\strand
(0,2) .. controls +(1.5,0) and +(1.5,0) ..
(0, 0) .. controls +(-1.5,0) and +(-1.5,0) ..
(0,2);
\end{knot}
\coordinate (p) at (current bounding box.center);
\end{tikzpicture}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
\begin{center}
\begin{tikzpicture}[baseline=(p), scale = 0.8]
\clip (-2,-1.7) rectangle + (4, 4);
\begin{knot}[
consider self intersections=true,
flip crossing = 2,
]
\strand
(0,2) .. controls +(2.2,0) and +(120:-2.2) ..
(210:2) .. controls +(120:2.2) and +(60:2.2) ..
(-30:2) .. controls +(60:-2.2) and +(-2.2,0) ..
(0,2);
\end{knot}
\coordinate (p) at (current bounding box.center);
\end{tikzpicture}
\end{center}
\end{minipage}
\end{center}
\vfill
\pagebreak
\problem{}
Below are the only four distinct knots with only one crossing. \par
Show that no nontrivial knot can have has fewer than three crossings. \par
\hint{There are 4 such knots. What are they?}
\begin{center}
\includegraphics[width=0.8\linewidth]{images/one crossing.png}
\end{center}
\begin{solution}
Draw all four. Each is isomorphic to the unknot.
\end{solution}
\vfill
\problem{}
Show that this is the unknot. \par
A wire or an extension cord may help.
\begin{center}
\includegraphics[width=0.35\linewidth]{images/big unknot.png}
\end{center}
\definition{}
As we said before, there are many ways to draw the same knot. \par
We call each drawing a \textit{projection}. Below are four projections of the \textit{figure-eight} knot.
\vspace{2mm}
\begin{center}
\includegraphics[width=0.8\linewidth]{images/figure eight.png}
\end{center}
\vspace{2mm}
\problem{}
Convince yourself that these are equivalent.
\vfill
\pagebreak

133
Advanced/Knots/parts/1 composition.tex Normal file
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@ -0,0 +1,133 @@
\section{Knot Composition}
Say we have two knots $A$ and $B$.
The knot $A \boxplus B$ is created by cutting $A$ and $B$ and joining their ends:
\begin{center}
\hfill
\begin{minipage}[t]{0.15\textwidth}
\begin{center}
\includegraphics[width=\linewidth]{images/composition a.png}
$A$
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{0.13\textwidth}
\begin{center}
\includegraphics[width=\linewidth]{images/composition b.png}
$B$
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{0.3\textwidth}
\begin{center}
\includegraphics[width=\linewidth]{images/composition c.png}
$A \boxplus B$
\end{center}
\end{minipage}
\hfill~
\end{center}
We must be careful to avoid new crossings when composing knots:
\vspace{2mm}
\begin{center}
\includegraphics[width=0.45\linewidth]{images/composition d.png}
\end{center}
\vspace{2mm}
We say a knot is \textit{composite} if it can be obtained by composing two other knots. \par
We say a knot is \textit{prime} otherwise.
\problem{}
For any knot $K$, what is $K \boxplus \text{unknot}$?
\vfill
\problem{}
Use a pencil or a cord to compose the figure-eight knot with itself.
\vfill
\vfill
\pagebreak{}
\problem{}
The following knots are composite. What are their prime components? \par
Try to make them with a cord! \par
\hint{Use the table at the back of this handout to decompose the second knot.}
\begin{center}
\hfill
\includegraphics[height=30mm]{images/decompose a.png}
\hfill
\includegraphics[height=30mm]{images/decompose b.png}
\hfill~\par
\vspace{4mm}
\end{center}
\begin{solution}
The first is easy, it's the trefoil composed with itself. \par
\vspace{2mm}
The second is knot $5_2$ composed with itself. \par
Note that the \say{three-crossing figure eight} is another projection of $5_2$. \par
The figure-eight knot is NOT a part of this composition. Look closely at its crossings.
\end{solution}
\vfill
\pagebreak
\definition{}
When we compose two knots, we may get different results. To fully understand this fact, we need to define knot \textit{orientation}.
\vspace{2mm}
An \textit{orientated knot} is created by defining a \say{direction of travel.} \par
There are two distinct ways to compose a pair of oriented knots:
\begin{center}
\hfill
\begin{minipage}[t]{0.25\textwidth}
\begin{center}
\includegraphics[width=\linewidth]{images/orientation b.png}
Matching orientation
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{0.25\textwidth}
\begin{center}
\includegraphics[width=\linewidth]{images/orientation c.png}
Inverse orientation
\end{center}
\end{minipage}
\hfill~
\end{center}
In this example, both compositions happen to give the same result. This is because the trefoil knot is \textit{invertible}: its direction can be reversed by deforming it. This is not true in general, as you will soon see.
\problem{}
Invert a directed trefoil.
\vfill
\problem{}
The smallest non-invertible knot is $8_{17}$, shown below. \par
Compose $8_{17}$ with itself to obtain two different knots.
\begin{center}
\includegraphics[height=30mm]{knot table/8_17.png} \par
\vspace{2mm}
{\large Knot $8_{17}$}
\end{center}
\begin{solution}
\begin{center}
\includegraphics[width=0.8\linewidth]{images/noninvertible.png}
\end{center}
\end{solution}
\vfill
\pagebreak

51
Advanced/Knots/tikzset.tex Normal file
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@ -0,0 +1,51 @@
\usetikzlibrary{
knots,
hobby,
decorations.pathreplacing,
shapes.geometric,
calc
}
\ifDebugKnot
\tikzset{
knot diagram/draft mode = crossings,
knot diagram/only when rendering/.style = {
show curve endpoints,
%show curve controls
}
}
\fi
\tikzset{
knot diagram/every strand/.append style={
line width = 0.8mm,
black
},
show curve controls/.style={
postaction=decorate,
decoration={
show path construction,
curveto code={
\draw[blue, dashed]
(\tikzinputsegmentfirst) -- (\tikzinputsegmentsupporta)
node [at end, draw, solid, red, inner sep=2pt]{}
;
\draw[blue, dashed]
(\tikzinputsegmentsupportb) -- (\tikzinputsegmentlast)
node [at start, draw, solid, red, inner sep=2pt]{}
node [at end, fill, red, ellipse, inner sep=2pt]{}
;
}
}
},
show curve endpoints/.style={
postaction=decorate,
decoration={
show path construction,
curveto code={
\node [fill, blue, ellipse, inner sep=2pt] at (\tikzinputsegmentlast) {};
}
}
}
}